Questions: Geometric Transformations: Translations

Question 1

Point A is at (3, −2). It is translated by vector (−5, 4). What are the coordinates of A'?

• A. (8, −6)
• B. (−2, 2)
• C. (−15, −8)
• D. (3, −2) — a translation doesn't change coordinates

Answer: (B)

Apply the rule (x + a, y + b): (3 + (−5), −2 + 4) = (−2, 2). Option A adds 5 instead of −5 — a sign error that is the most common mistake. Option C multiplies instead of adds, confusing translation with dilation. Option D misunderstands what a translation does. A translation always adds the vector components; it never multiplies or leaves coordinates unchanged (unless the vector is (0, 0)).

Question 2

A triangle with side lengths 3, 4, and 5 is translated 20 units to the left. What are the side lengths of the image?

• A. 3, 4, and 5 — translations preserve all distances
• B. 23, 24, and 25 — the translation distance is added to each side
• C. −17, −16, and −15 — moving left subtracts from the measurements
• D. It depends on the orientation of the triangle

Answer: (A)

Translations are isometries — they preserve all distances. Side lengths are unchanged regardless of how far or in what direction the figure moves. Only dilations change size. The translation vector affects position, not shape or size. A common misconception conflates moving a figure with scaling it.

Question 3

Two consecutive translations — first by vector (3, −1) and then by (−7, 4) — produce the same result as a single translation by (−4, 3).

• T. True
• F. False

Answer: True

Translations compose by vector addition: (3 + (−7), −1 + 4) = (−4, 3). The combined effect is always another translation. This works because each translation applies the same constant addition to every point, and additions accumulate linearly. The set of all translations forms a group under composition — a structure where combining two elements always yields another element of the same type.

Question 4

Translating a figure can change its orientation — for example, a clockwise-oriented triangle may become counterclockwise-oriented after translation.

• T. True
• F. False

Answer: False

Translations preserve orientation. Every point moves by the same displacement, so there is no flipping or rotating. Clockwise stays clockwise; counterclockwise stays counterclockwise. Orientation reversal is caused by reflections (which flip the figure), not translations. This is one of the key distinctions between translation and reflection as types of rigid motions.

Question 5

Explain why a translation is called an isometry, and what this guarantees about the relationship between a figure and its translated image.

Model answer: An isometry is a transformation that preserves all distances. In a translation by vector (a, b), every point (x, y) maps to (x+a, y+b). If two points P and Q are distance d apart, their images P' and Q' are still distance d apart — the added constants (a, b) cancel when you compute the distance between them. Since all distances are preserved, angle measures are also preserved (angles depend only on distances between three points). The image is therefore congruent to the preimage: same shape, same size, different location.

Congruence is the practical payoff of isometry. Whenever you need to prove two figures are congruent in geometry, demonstrating they are related by a translation (or other isometry) is a valid proof strategy. The isometry property is also what distinguishes translations from dilations, which scale distances and produce similar but not congruent images.